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democrite-00023623, version 1 - 21 Jan 2005
Dirac’s inspired point form and hadron form factors
B. Desplanquesa ∗aLaboratoire de Physique Subatomique et de Cosmologie
(UMR CNRS/IN2P3-UJF-INPG), F-38026 Grenoble Cedex, France
Noticing that the “point-form” approach referred to in many recent works implies physics described on hyperplanes, an approach inspired from Dirac’s one, which involves a hyperboloid surface, is presented. A few features pertinent to this new approach are emphasized. Consequences as for the calculation of form factors are discussed.
1. INTRODUCTION
Looking at a Hamiltonian formulation of relativistic dynamics, Dirac was led to consider various forms, depending on the symmetry properties of the hypersurface that is chosen in this order [ 1]. Accordingly, the generators of the Poincar´e algebra drop into dynamical or kinematic ones. Among the different forms, the point-form approach, which is based on a hyperboloid surface, t2−~x2 = τ , is probably the most aesthetic one in the sense that
the space-time displacement operators, Pµ, are the only ones to contain the interaction
while the boost and rotation operators, Mµ ν altogether, have a kinematic character. This
approach is also the less known one, perhaps because dealing with a hyperboloid surface is not so easy as working with the hyperplanes that underly the other forms (instant and front). It nevertheless received some attention recently within the framework of relativistic quantum mechanics (RQM). Due to the kinematic character of boosts, its application to the calculation of form factors can be easily performed a priori and, moreover, these quantities generally evidence the important property of being Lorentz invariant.
A “point-form” (“P.F.”) approach has been successfully used for the calculation of the nucleon form factors [ 2]. It however fails in reproducing the form factor of much simpler systems, including the pion [ 3, 4, 5]. The asymptotic behavior is missed and the drop-off at small Q2 is too fast in the case of a strongly-bound system. Analyzing the
results, it was found that this “point-form”, where the dynamical or kinematic character of the Poincar´e generators is the same as for Dirac’s one, implies physics described on hyperplanes [ 6]. This approach is nothing but that one presented by Bakamjian as being an “instant form · · · which displays the symmetry properties inherently present in the point form” [ 7]. Sokolov mentioned it was involving “hyperplanes orthogonal · · · to the 4-velocity · · · of the system” under consideration, adding it was not identical to the point form proposed by Dirac [ 8]. Developing an approach more in the spirit of the original one therefore remains to be made.
2. FORMALISM: A FEW INGREDIENTS
Each RQM approach is characterized by the relation that the momenta of a system and its constituents fulfill off-energy shell. This one is determined by the symmetry properties of the hypersurface which the physics is formulated on. In absence of particular direction on a hyperboloid-type surface, it necessarily takes the form of a Lorentz scalar. One should also recover the momentum conservation in the non-relativistic limit, P
ip~i = ~P.
Thus, for the two-body system we are considering here, the expected relation could read: (p1+ p2− P )2 = ( ~p1+ ~p2− ~P)2− (e1+ e2− EP)2 = 0 . (1)
Such a constraint is obtained from integrating plane waves on the hypersurface, x2 = 0:
Z
d4x ei(p−p′)·x δ(x2) ǫ(U ·x) = 4iπ2 δ((p − p′)2) ǫ(U ·(p − p′)) , (2)
where pµand p′µare replaced by (p
1+p2)µand Pµ, and Uµsatisfies U2 ≥ 0. To understand
the ingredients entering the l.h.s. of the above equation, the “time” evolution should be examined [ 9]. This goes beyond considering the upper part of a hyperboloid surface often mentioned in the literature. Interestingly, Eq. (1) can be cast into the following form: ( ~p1+ ~p2− ~P) = ˆu(e1+ e2− EP) (ˆu2 = 1, ˆunot fixed) , (3)
which is very similar to a front-form one, but the unit vector, ˆu, has no fixed direction. The next step consists in considering a wave equation, which can be obtained from taking the square of the momentum operator, Pµ:
M2− p2ΦP(~p1, ~p2) = − Z Z d~p′ 1 (2 π)3 d~p′ 2 (2 π)3 1 (2 e1 2 e2 2 e′1 2 e′2)1/2 × (p + p′)· ∂ p−p′ 4 π2 δ (p − p′)2 ǫU·(p − p′) ! 4 m2 g2 µ2+ · · · ΦP(~p ′ 1, ~p′2) , (4) where pµ = pµ
1 + pµ2. One should determine under which conditions it admits solutions
verifying Eq. (1) and, at the same time, leads to a relevant mass operator. With this aim, we assume Uµ ∝ c(p − P )µ+ c′(p′− P )µ, from which we get:
~ U U0 = ˆu= ~ p− ~P e− EP = ~p ′− ~P e′ − E P = ~p ′′− ~P e′′− E P = · · · . (5)
to take into account higher-order meson-exchange contributions, are actually well known as part of the general construction of the Poincar´e algebra in RQM approaches [ 11].
The present point form implies that the system described in this way evidences a new degree of freedom. In the c.m., a zero total momentum is obtained by adding the individual contributions of constituents and an interaction one, consistently with the fact that ~P contains the interaction. The configuration so obtained points isotropically to all directions as sketched in Fig. 2 of Ref. [ 9]. This new degree of freedom appears explicitly in the definition of the norm, beside the integration on the internal ~k variable:
N = Z d~k (2π)3 φ 2 0(k) Z d~u 2 π δ(1 − ~u 2) M2 (u·P )2 , (6)
where φ0(k) represents a solution of a mass operator. Another aspect of the present point
form concerns the velocity operator, ~V, entering the construction of the Poincar´e algebra, and the corresponding ~P. Their expressions, which differ from earlier ones, read:
~ V = 1 M ~ p+ ~u 4 m ˜V 2 u·P , ~P = ~p + ~u 4 m ˜V 2 u·P earlier “P.F.” : ~V = ~p 2 ek , ~P = M 2 ek ~ p. (7) Despite unusual features, the present point form can be consistently developed. It evi-dences similarities with the instant and front forms in that the hypersurface it is formu-lated on is independent of the system under consideration, which is at the origin of the above constraints. These ones are absent in the earlier “point form” where the kinematic character of boosts is trivial, the operation affecting both the system and the hyperplane used for its description, their respective velocity and orientation being related.
3. SOLVED AND UNSOLVED PROBLEMS, DISCUSSION, OUTLOOK For a part, the present work was motivated by the drawbacks that an earlier point form evidences for the form factors of strongly-bound systems calculated in the single-particle current approximation [ 3, 4, 12]. Some results are presented in Fig. 1 for the pion charge form factor. At high Q2, the new point form (D.P.F.) shows a Q−4 behavior, like the
instant- and front-form results, while the earlier point form (“P.F.”) is providing a Q−8
one. The change in the power law is largely due to the form of the velocity operator, Eq. (7), which, containing some dependence on ˆu, makes less difficult to match the initial- and final-state momenta with those of the struck constituent. At low Q2, the new
point form does better than the earlier one but the improvement is not impressive. More important however, the bad behavior is shared by results obtained in the instant and front forms with parallel kinematics (I.F.+F.F.(parallel)) [ 12]. All of them evidence a charge squared radius scaling like the inverse of the squared mass of the system. In comparison, the standard instant and front forms (I.F. (Breit frame) and F.F. (perp.) in Fig. 1) do well.
transla-0 0.05 0.1 0.15 0.2 Q2 [(GeV/c)2] 0 0.2 0.4 0.6 0.8 1 F 1 (Q 2 ) F.F. (perp.) I.F. (Breit frame) D.P.F. I.F.+F.F. (parallel) "P.F." m=0.3 GeV, M=0.14 GeV 0 2 4 6 8 10 Q2 [ (GeV/c)2] 10-6 10-4 10-2 100 Q 2 F 1 (Q 2 ) [(GeV/c) 2 ] F.F. (perp.) I.F. (Breit frame) D.P.F.
I.F.+F.F. (parallel) "P.F."
m=0.3 GeV, M=0.14 GeV
Figure 1. Pion charge form factor in different forms of relativistic quantum mechanics.
This relation [ 13] cannot be verified exactly at the operator level in RQM approaches with a single-particle current but one can require it is verified, at least, at the matrix-element level. With this respect, what is an advantage for the point-form approach becomes a disadvantage as there is no frame where one can minimize the effect of a violation of the above relation (a factor 2-3 for the nucleon and roughly 6 for the pion). On the contrary, in the instant and front forms, one can consider different frames. It turns out that the instant- and front-form results for a perpendicular kinematics (standard ones) verify the above equality while those for a parallel kinematics do badly, similarly to the point-form case. It thus appears that Poincar´e space-time translation invariance could be more important than the Lorentz one and that the intrinsic Lorentz covariance of the point-form approach is not so much an advantage than what could be a priori expected. REFERENCES
1. P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392.
2. R.F. Wagenbrunn, et al., Phys. Lett. B 511 (2001) 33.
3. B. Desplanques and L. Theußl, Eur. Phys. J. A13 (2002) 461.
4. A. Amghar, B. Desplanques and L. Theußl, Nucl. Phys. A 714 (2003) 213. 5. A. Amghar, B. Desplanques, L. Theußl, Phys. Lett. B 574 (2003) 201. 6. B. Desplanques, S. Noguera, L. Theußl, Phys. Rev. C 65 (2002) 038202. 7. B. Bakamjian, Phys. Rev. 121 (1961) 1849.
8. S.N. Sokolov, Theor. Math. Phys. 62 (1985) 140. 9. B. Desplanques, Nucl. Phys. A 748 (2005) 139.
10. B. Bakamjian, L.H. Thomas, Phys. Rev. 92 (1953) 1300. 11. B. Keister, W. Polyzou, Adv. Nucl. Phys. 20 (1991) 225. 12. B. Desplanques, preprint, nucl-th/0407074.
